Directional Recurrence for Infinite Measure Preserving Z^d Actions

TL;DR

This paper introduces the concept of directional recurrence in infinite measure preserving Z^d actions, establishes its equivalence through different definitions, and explores the structure and examples of recurrent directions.

Contribution

It provides a new intrinsic and flow-based definition of directional recurrence, proves their equivalence, and constructs novel examples addressing open questions.

Findings

The set of recurrent directions is always a G_delta set.

Existence of recurrent actions with no recurrent directions.

Recurrent actions can lack recurrence in irrational directions despite sub-actions being recurrent.

Abstract

We define directional recurrence for infinite measure preserving Z^d actions both intrinsically and via the unit suspension flow and prove that the two definitions are equivalent. We study the structure of the set of recurrent directions and show it is always a G_delta set. We construct an example of a recurrent action with no recurrent directions, answering a question posed in a 2007 paper of Daniel J. Rudolph. We also show by example that it is possible for a recurrent action to not be recurrent in an irrational direction even if all its sub-actions are recurrent.